The number is 105263157894736842. Hey, I said it was quite large, didn't I?

Start by realizing that the second to last digit of the original number is 4. (This is because the new number's last digit has to be double the old last digit which was two. And the new last digit is the original next to last digit - everything gets shifted when you move the 2 to the front).

Next write down:

..42
+ ..42
= ...4

(This is because you are doubling the first number to get the new one.) It will become obvious that the second to last digit of the new number (and third to last of the new one) is 8:

...842
+ ...842
= ....84

Note that for the next step, 8+8 will be 16, and so we will use the 6 as the third to last digit for the new number, but remember to carry over the 10 for the next time, when 6+6+1 will yield 13:

...6842
+ ...6842
= ...3684

If you keep this up long enough (don't get discouraged, but be careful in your arithmetics) you will eventually get a valid equasion: